For what values of \( x \) and \( y \) , are the following matrices equal?
\( A=\begin{bmatrix}2x+1 & 3y\\0 & y^2-5y\end{bmatrix},\ B=\begin{bmatrix}x+3 & y^2+2\\0 & -6\end{bmatrix} \)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( x=2,y=2 \)
Equal matrices have equal corresponding entries.
(1, 1) entries:
\( 2x+1=x+3 \), so x = 2
(1, 2) entries:
\( 3y=y^2+2 \)
\( (y-1)(y-2)=0 \), so y = 1 or 2
(2, 2) entries:
\( y^2-5y=-6 \)
\( (y-2)(y-3)=0 \), so y = 2 or 3
Both conditions on y must hold, so y = 2.
Why the other options are incorrect
Option (2) \( x=2,y=1 \) — Incorrect. y = 1 satisfies only the (1, 2) entry; in the (2, 2) entry \( 1-5=-4 \), not −6.
Option (3) \( x=1,y=2 \) — Incorrect. With x = 1 the (1, 1) entries are 3 and 4, which are not equal.
Option (4) \( x=2,y=3 \) — Incorrect. y = 3 satisfies only the (2, 2) entry; in the (1, 2) entry 3y = 9 but \( y^2+2=11 \).